Fast matrix multiplication techniques based on the Adleman-Lipton model

TL;DR

This paper explores encoding fast matrix multiplication algorithms, specifically Strassen's, into DNA computing using the Adleman-Lipton model, demonstrating the potential for scalable DNA-based matrix computations and their broader applications.

Contribution

It presents a theoretical framework for implementing all fast matrix multiplication algorithms on DNA computers using the residue number system.

Findings

DNA encoding of Strassen's algorithm demonstrated

Scalable DNA computing model proposed

Potential for DNA-based matrix operations discussed

Abstract

On distributed memory electronic computers, the implementation and association of fast parallel matrix multiplication algorithms has yielded astounding results and insights. In this discourse, we use the tools of molecular biology to demonstrate the theoretical encoding of Strassen's fast matrix multiplication algorithm with DNA based on an $n$-moduli set in the residue number system, thereby demonstrating the viability of computational mathematics with DNA. As a result, a general scalable implementation of this model in the DNA computing paradigm is presented and can be generalized to the application of \emph{all} fast matrix multiplication algorithms on a DNA computer. We also discuss the practical capabilities and issues of this scalable implementation. Fast methods of matrix computations with DNA are important because they also allow for the efficient implementation of other algorithms (i.e. inversion, computing determinants, and graph theory) with DNA.